Theorem 4.1 ([1]}, Chapter \(V.3\) ) 
Consider a closed hyperbolic three-manifold (or more generally any manifold with Ricci curvature \(\ge -2\) ). Then
\(\mathfrak {I}_Y\ge \frac{1}{64\pi ^5}\left(\frac{\mathrm {vol}(Y)}{\cosh (d(Y))-1}\right)^4\) 
